Derivation-based Noncommutative Field Theories on $AF$ algebras

Abstract

In this paper, we start the investigation of a new natural approach to "unifying" noncommutative gauge field theories (NCGFT) based on approximately finite-dimensional ($AF$) $C^{\ast }$-algebras. The defining inductive sequence of an $AF$ $C^{\ast }$-algebra is lifted to enable the construction of a sequence of NCGFT of Yang-Mills-Higgs types. The present paper focus on derivation-based noncommutative field theories. A mathematical study of the ingredients involved in the construction of a NCGFT is given in the framework of $AF$ $C^{\ast }$-algebras: derivation-based differential calculus, modules, connections, metrics and Hodge $\star$-operators, Lagrangians... Some physical applications concerning mass spectra generated by Spontaneous Symmetry Breaking Mechanisms (SSBM) are proposed using numerical computations for specific situations.

42 pages, 20 figures

References in corpus (2)

• Formulation of gauge theories on transitive Lie algebroids
• Noncommutative generalization of SU(n)-principal fiber bundles: a review

Cited by in corpus (1)

• Lifting Bratteli Diagrams between Krajewski Diagrams: Spectral Triples, Spectral Actions, and $AF$ algebras